${\left({\text{In}}_x{\text{Ga}}_{1-x}\right)}_2{\text{O}}_3$ alloys for transparent electronics

${\left({\text{In}}_x{\text{Ga}}_{1-x}\right)}_2{\text{O}}_3$ alloys show promise as transparent conducting oxides. Using hybrid density functional calculations, band gaps, formation enthalpies, and structural parameters are determined for monoclinic and bixbyite crystal structures. In the monoclinic phase the band gap exhibits a linear dependence on alloy concentration, whereas in the bixbyite phase a large band-gap bowing occurs. The calculated formation enthalpies show that the monoclinic structure is favorable for In compositions up to 50% and bixbyite for larger compositions. This is caused by In strongly preferring sixfold oxygen coordination. The formation enthalpy of the 50:50 monoclinic alloy is much lower than the formation enthalpy of the 50:50 bixbyite alloy and also lower than most monoclinic alloys with lower In concentration; these trends are explained in terms of local strain. Consequences for experiment and applications are discussed.

Figure 1

(a) Unit cell of bixbyite ${\text{In}}_2{\text{O}}_3$, where the symmetric octahedral sites are dark gray and the distorted site is light blue. (b) A $1\times 1\times 2$ supercell of monoclinic ${\text{Ga}}_2{\text{O}}_3$, with polyhedra indicating the tetrahedral (dark brown) and octahedral (light silver) sites [13].

Figure 2

Band alignment between ${\text{Ga}}_2{\text{O}}_3$ and ${\text{In}}_2{\text{O}}_3$ in monoclinic and bixbyite structures. The energy zero is the vacuum level.

Figure 3

Pseudocubic lattice parameter of the lowest-energy ordered ${\left({\text{In}}_x{\text{Ga}}_{1-x}\right)}_2{\text{O}}_3$ alloys in bixbyite (blue squares) and monoclinic (red circles) structures. The orange and turquoise lines indicate Vegard's law.

Figure 4

Alloy formation enthalpy [Eq. (1)] of ordered ${\left({\text{In}}_x{\text{Ga}}_{1-x}\right)}_2{\text{O}}_3$ alloys in the bixbyite (blue squares) and monoclinic (red solid circles) structure. The red open circles are higher-energy configurations, as discussed in the text. The parabolas indicate lower limits within the model of Eq. (2), with $\mathrm{\Delta }{H}_0=35$ meV (orange) and 94 meV (turquoise).

Figure 5

Direct band gap at $\mathrm{\Gamma }$ for ordered ${\left({\text{In}}_x{\text{Ga}}_{1-x}\right)}_2{\text{O}}_3$ alloys. (a) For the monoclinic ${\text{Ga}}_2{\text{O}}_3$ structure (red circles), the orange line is a linear interpolation between the binary compounds. (b) In the bixbyite structure, the blue open squares denote the band gap corresponding to the average of the three highest valence bands; this gap is dark. The lines are the parabolic fit based on Eq. (4). The green solid squares denote the lowest direct transition at $\mathrm{\Gamma }$ with a large dipole matrix element. Experimental values (triangles) are from Refs. [15] (downward open triangles), [19] (downward solid triangles), [24] (upward open triangle), and [20] (upward solid triangles).